论文标题
打开和封闭曲线的多项式
Knot polynomials of open and closed curves
论文作者
论文摘要
在本手稿中，我们引入了一种测量3空间曲线纠缠的方法，该方法将结和将多项式链接到打开曲线的概念。我们在3空间中定义了曲线的括号多项式，并表明它具有实际系数，并且是链坐标的连续函数。这用于定义琼斯多项式，以适用于3空间的开放曲线和封闭曲线。对于开放曲线，它具有实际系数，并且是链坐标的连续函数，并且随着曲线的端点倾向于重合，琼斯的开放曲线多项式倾向于所产生的结的曲线。对于封闭的曲线，它是拓扑不变的，作为经典的琼斯多项式。我们展示了这些措施如何获得多边形链的更简单的表达，并在3和4边的链中为其计算提供了有限的形式。
In this manuscript we introduce a method to measure entanglement of curves in 3-space that extends the notion of knot and link polynomials to open curves. We define the bracket polynomial of curves in 3-space and show that it has real coefficients and is a continuous function of the chain coordinates. This is used to define the Jones polynomial in a way that it is applicable to both open and closed curves in 3-space. For open curves, it has real coefficients and it is a continuous function of the chain coordinates and as the endpoints of the curve tend to coincide, the Jones polynomial of the open curve tends to that of the resulting knot. For closed curves, it is a topological invariant, as the classical Jones polynomial. We show how these measures attain a simpler expression for polygonal chains and provide a finite form for their computation in the case of chains of 3 and 4 edges.
